Cogito
Algebra 1 · Chapter 8 · Lesson 2
Completing the Square and the Formula
Where the formula comes from.
10 problems · about 22 minutes · A-REI.B.4, A-SSE.B.3
What this lesson teaches
The student completes the square, derives the quadratic formula, and uses the discriminant.
- Completing the square rewrites a quadratic in vertex form.
- The quadratic formula is what completing the square gives in general.
- The discriminant b² − 4ac decides how many real roots there are.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Parabolas and Their Roots: x² − 7x + 12 = 0. What is the larger root?
Answer 4
Why x = 4.
Review — Factoring Quadratic Trinomials: Multiply to 18, add to 9. Larger number?
Answer 6
Why 6.
Warm Up
Straightforward practice. Get the method working first.
4 problemsx² − 2x − 8. What is b² − 4ac?
Answer 36
Why 36, so two roots.
A negative discriminant means what?
Answer No real roots; the parabola misses the axis.
Why No real roots.
x² + 8x + ? completes the square. What number?
Answer 16
Why Half of 8, squared.
x² + 10x + ? completes the square. What number?
Answer 25
Why Half of 10, squared.
Build It Up
The same ideas with more to keep track of.
2 problemsx² − 6x + 9. What is b² − 4ac?
Answer 0
Why 36 − 36.
x² + 3x + 1. What is b² − 4ac?
Answer 5
Why 9 − 4.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Discriminant: x² − 4x + 3. What is b² − 4ac?
Answer 4
Why 4, which is positive, so two roots.
How Many Roots: x² + 2x + 5. What is b² − 4ac?
Answer -16
Why −16, which is negative, so no real roots.